Now, consider the disproportionate growth asymptotic. The convergences which were previously so helpful as \(\beta _n \rightarrow \beta > 0\)  are not useful as \(\beta _n \rightarrow 0\) . The limits of \(F_n\)  and \(\hspace{0.83328pt}\overline{\hspace{-0.83328pt}M\hspace{-0.83328pt}}\hspace{0.83328pt}_n(z)\)  are degenerate, as all noise eigenvalues converge to one. The upper edge of the support of \(F_n\)  lies approximately at \((1+\sqrt{\beta _n})^2\) . Theorem REF  tells us the leading eigenvalues of \(\widetilde{S}_n\)  may still emerge from the bulk, but only by a multiple of \(\sqrt{\beta _n}\) . The phenomena of interest are no longer made visible by taking limits, but must be captured by detailed finite-\(n\)  analysis. This will require bounds on the convergence rate of \(s_n(z)\)  to \(\bar{s}_n(z)\) , developed in Section , as well as careful tracking of the precise size of remainders, where previously it sufficed to know that such remainders tended to zero. The proof of Theorem REF  is a refinement of the proofs of Theorem 2.9 of [1]} and Lemma 6.1 of [2]}. The proof of Theorem REF  is a refinement of Theorem 2.10 of [1]}.
